Vestigial structures give us evidence of organs or body parts that existed at one time and eventually were not needed by the species to survive. For example, you have an appendix but it's not an organ that is used very often, if at all, by the body. However, in rabbits the appendix is a vital part of their digestive system. Scientists think that the appendix helps to digest large amounts of wild grasses and grains, which would have been a useful organ to early humans while they were relying heavily on those fibrous foods to sustain themselves. So vestigial structures basically give us an idea of something that a species needed at one time in their past to survive but due to change and evolution, they don't need it for survival anymore so it gradually has started to disappear.
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